#8633648

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#41816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(18,22)(19,25)(24,30)(27,28)(33,34,38,41)(35,39,43,37)(36,44,47,40)(42,46,45,48);(1,2)(3,8)(4,5)(6,7)(9,13)(10,14)(11,16)(12,15)(17,18,21,22)(19,26,25,23)(20,24,29,30)(27,32,28,31)(33,35)(34,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(2,7)(4,13)(8,15)(10,16)(17,18,21,22)(19,23,25,26)(20,24,29,30)(27,31,28,32)(33,36,38,47)(34,40,41,44)(35,42,43,45)(37,46,39,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,15)(9,11)(13,16)(17,19)(18,23)(20,28)(21,25)(22,26)(24,32)(27,29)(30,31)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,45)(47,48);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45);(1,5,6,9)(2,4,7,13)(3,11,12,14)(8,16,15,10)(17,20,21,29)(18,24,22,30)(19,27,25,28)(23,31,26,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540844);